Vectorial point spread function and optical transfer function in oblique plane imaging
Jeongmin Kim,1,3 Tongcang Li,2,3 Yuan Wang,2,3 and Xiang Zhang1,2,3,* 1Department of Mechanical Engineering, University of California, Berkeley, California 94720, USA 2NSF Nanoscale Science and Engineering Center, 3112 Etcheverry Hall, University of California, Berkeley, California 94720, USA 3Materials Science Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, California 94720, USA *[email protected]
Abstract: Oblique plane imaging, using remote focusing with a tilted mirror, enables direct two-dimensional (2D) imaging of any inclined plane of interest in three-dimensional (3D) specimens. It can image real-time dynamics of a living sample that changes rapidly or evolves its structure along arbitrary orientations. It also allows direct observations of any tilted target plane in an object of which orientational information is inaccessible during sample preparation. In this work, we study the optical resolution of this innovative wide-field imaging method. Using the vectorial diffraction theory, we formulate the vectorial point spread function (PSF) of direct oblique plane imaging. The anisotropic lateral resolving power caused by light clipping from the tilted mirror is theoretically analyzed for all oblique angles. We show that the 2D PSF in oblique plane imaging is conceptually different from the inclined 2D slice of the 3D PSF in conventional lateral imaging. Vectorial optical transfer function (OTF) of oblique plane imaging is also calculated by the fast Fourier transform (FFT) method to study effects of oblique angles on frequency responses. © 2014 Optical Society of America OCIS codes: (260.1960) Diffraction theory; (180.0180) Microscopy; (110.0110) Imaging systems; (110.4850) Optical transfer functions. References and links 1. F. Anselmi, C. Ventalon, A. Bègue, D. Ogden, and V. Emiliani, “Three-dimensional imaging and photostimulation by remote-focusing and holographic light patterning,” Proc. Natl. Acad. Sci. U.S.A. 108(49), 19504–19509 (2011). 2. C. W. Smith, E. J. Botcherby, M. J. Booth, R. Juškaitis, and T. Wilson, “Agitation-free multiphoton microscopy of oblique planes,” Opt. Lett. 36(5), 663–665 (2011). 3. C. W. Smith, E. J. Botcherby, and T. Wilson, “Resolution of oblique-plane images in sectioning microscopy,” Opt. Express 19(3), 2662–2669 (2011). 4. W. Göbel and F. Helmchen, “New angles on neuronal dendrites in vivo,” J. Neurophysiol. 98(6), 3770–3779 (2007). 5. C. Dunsby, “Optically sectioned imaging by oblique plane microscopy,” Opt. Express 16(25), 20306–20316 (2008). 6. S. Kumar, D. Wilding, M. B. Sikkel, A. R. Lyon, K. T. MacLeod, and C. Dunsby, “High-speed 2D and 3D fluorescence microscopy of cardiac myocytes,” Opt. Express 19(15), 13839–13847 (2011). 7. F. Cutrale and E. Gratton, “Inclined selective plane illumination microscopy adaptor for conventional microscopes,” Microsc. Res. Tech. 75(11), 1461–1466 (2012). 8. E. J. Botcherby, R. Juskaitis, M. J. Booth, and T. Wilson, “Aberration-free optical refocusing in high numerical aperture microscopy,” Opt. Lett. 32(14), 2007–2009 (2007). 9. E. J. Botcherby, R. Juskaitis, M. J. Booth, and T. Wilson, “An optical technique for remote focusing in microscopy,” Opt. Commun. 281(4), 880–887 (2008). 10. C. J. R. Sheppard and H. J. Matthews, “Imaging in high-aperture optical systems,” J. Opt. Soc. Am. A 4(8), 1354–1360 (1987). 11. M. Gu, Advanced Optical Imaging Theory (Springer, 2000). 12. B. Richards and E. Wolf, “Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic System,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
#207115 - $15.00 USD Received 25 Feb 2014; revised 8 Apr 2014; accepted 8 Apr 2014; published 1 May 2014 (C) 2014 OSA 5 May 2014 | Vol. 22, No. 9 | DOI:10.1364/OE.22.011140 | OPTICS EXPRESS 11140 13. J. D. Jackson, Classical Electrodynamics (Wiley, 1999). 14. J. A. Stratton and L. J. Chu, “Diffraction Theory of Electromagnetic Waves,” Phys. Rev. 56(1), 99–107 (1939). 15. J. J. Stamnes, Waves in Focal Regions: Propagation, Diffraction, and Focusing of Light, Sound, and Water Waves (Adam Hilger, 1986). 16. T. D. Visser and S. H. Wiersma, “Spherical aberration and the electromagnetic field in high-aperture systems,” J. Opt. Soc. Am. A 8(9), 1404–1410 (1991). 17. C. J. R. Sheppard, A. Choudhury, and J. Gannaway, “Electromagnetic field near focus of wide-angular lens and mirror systems,” IEEE J. Microwave Opt. Acoust. 1(4), 129–132 (1977). 18. E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39(4), 205–210 (1981). 19. Y. J. Li, “Focal shifts in diffracted converging electromagnetic waves. I. Kirchhoff theory,” J. Opt. Soc. Am. A 22(1), 68–76 (2005). 20. Y. J. Li, “Focal shifts in diffracted converging electromagnetic waves. II. Rayleigh theory,” J. Opt. Soc. Am. A 22(1), 77–83 (2005). 21. E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 349–357 (1959). 22. C. J. R. Sheppard, M. Gu, Y. Kawata, and S. Kawata, “Three-dimensional transfer functions for high-aperture systems,” J. Opt. Soc. Am. A 11(2), 593–598 (1994). 23. B. Frieden, “Optical Transfer of the Three-Dimensional Object,” J. Opt. Soc. Am. 57(1), 56–65 (1967). 24. C. J. R. Sheppard and K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik (Stuttg.) 107, 79–87 (1997). 25. M. R. Arnison and C. J. R. Sheppard, “A 3D vectorial optical transfer function suitable for arbitrary pupil functions,” Opt. Commun. 211(1-6), 53–63 (2002). 1. Introduction Oblique plane microscopy (OPM) [1–7] images 2D cross-sections of a specimen that are tilted from the focal plane of a microscope objective lens. This microscopy has certain advantages in circumstances where an interested plane of a sample is inclined to conventional microscope’s image plane [2–4], or biological processes of a living sample involve rapid changes in its structural orientation [2]. Compared to 3D scanning microscopy that typically extracts oblique planar information from slow 3D measurements, OPM provides a high- speed, cost-effective imaging method. Commercial objective lenses satisfying the Abbe’s sine condition suffer from optical aberrations if they are used to image any part of objects lying out of their focal planes. Remote focusing [8,9], which uses a second objective lens to cancel out these optical aberrations, provides a solution to form an aberration-free, 3D intermediate image of an out- of-focus object. Oblique plane imaging of this intermediate image is then conceptually feasible by employing another microscope with an inclined angle at the expense of partial use of its numerical aperture (NA) for detection, leading to resolution loss in an anisotropic manner. Dunsby [5] has estimated the imaging resolution of oblique plane microscopy by approximating the full-width at half-maximum (FWHM) of the PSF from an effective NA concept, which is inaccurate and does not provide an analytical clue on non-isotropic lateral resolution. Anselmi et al. [1] proposed a wide-field oblique plane imaging method by a remote tilting technique, which has simpler configuration than Dunsby’s. They explained qualitatively two mechanisms of resolution loss due to the possible light clipping and the inclined detection PSF. Because their experimental oblique angle was explored only up to 14° where such effect is small, no theoretical study on resolution was reported. On the other hand, Smith et al. [2,3] showed point-scanning oblique plane microscopy using a remote scanning technique. They studied non-isotropic lateral resolution for all oblique angles in terms of spatial cutoff frequencies deduced from the region of support for the 3D OTF in Fourier space. However, their analysis for point-scanning microscopy is not applicable to wide-field oblique plane imaging because their system has no light clipping with a different overall PSF. Accurate theoretical resolution of direct oblique plane imaging is thus still unclear. In this paper, we theoretically calculate the accurate optical resolution of the wide-field oblique plane imaging for the first time. We derive a mathematical expression of the 3D pupil
#207115 - $15.00 USD Received 25 Feb 2014; revised 8 Apr 2014; accepted 8 Apr 2014; published 1 May 2014 (C) 2014 OSA 5 May 2014 | Vol. 22, No. 9 | DOI:10.1364/OE.22.011140 | OPTICS EXPRESS 11141 function influenced by light clipping in any oblique angle between 0° and 90°. Then we calculate 2D intensity PSFs based on the vectorial diffraction theory that gives accurate results even for high NA systems (NA>0.6). FWHMs of the PSFs with different oblique angles are calculated to characterize the lateral resolution. We also calculate vectorial OTF from the FFT of the PSF to examine the effects of oblique angles on the spatial cutoff frequency. 2. Schematic of the oblique plane imaging An oblique plane imaging system [1] is schematically shown in Fig. 1. The back focal planes of the two aplanatic objective lenses are relayed back-to-back by the L1-L2 optics. This layout compensates for aberrated optical wavefronts from the out-of-focus object near focal regions by the odd parity condition [8,9], thereby extending the depth of field [9]. Thus for an object lying within this range, the diffraction-limited 3D replica is formed in the remote space with a 3D isotropic magnification of the ratio of object/remote medium indices. The OBJ2-L3 constitutes another microscope to capture the oblique plane image of the remote object. Figure 1(a-c) shows that the α-tilted plane (xαyα) in the object space is optically conjugate with the detector plane in the image space due to the α/2-tilted mirror in the remote space. The pink beam in Fig. 1 shows how light is clipped at the OBJ2 induced by the tilted mirror. This light loss leads to a partial use of the OBJ2’s exit pupil (the blue arc) and this rotationally asymmetric pupil yields an anisotropic resolving power. The NA or half-cone angle of the OBJ2 should be chosen greater than the mirror tilt angle, α/2, to prevent a complete loss of light from detection. For example, an axial plane imaging (α = 90°) requires NA greater than 0.71 in air medium. In general, the use of as high NA as possible is desirable to minimize the clipping of the signal light.
Fig. 1. Conceptual diagram of oblique plane imaging. OBJ: objective lens (EP: exit pupil; BFP: back focal plane); BS: beam splitter; L: lens; M: mirror. The beam path for an on-axis point object to the detector is shown in green, while the light clipping at the OBJ2 induced by the tilted mirror is illustrated in pink. Coordinates at (a) object space: an oblique plane (xαyα)
inclined by α with respect to the focal plane (xy) of the OBJ1, (b) remote space: the xy11 image
plane conjugate with the xαyα plane is rotated back to the xy11'' plane (the focal plane of the OBJ2) by the α/2-tilted mirror, and (c) image space: the lateral detection plane ( xy22) is conjugate with the xαyα object plane. Rays (green, light blue) from two points on the oblique
xαyα plane are focused on the xy22 plane. We note that the optical arrangement in Fig. 1 enables a 2D imaging of any oblique plane by controlling the tip-tilt of the small mirror M, of which size is about the working distance of the objective. In addition to this tip-tilt, a 3D translation of the mirror could also be realized to properly shift the image plane with neither specimen agitations nor additional optical
#207115 - $15.00 USD Received 25 Feb 2014; revised 8 Apr 2014; accepted 8 Apr 2014; published 1 May 2014 (C) 2014 OSA 5 May 2014 | Vol. 22, No. 9 | DOI:10.1364/OE.22.011140 | OPTICS EXPRESS 11142 aberrations induced. Furthermore, the PSF in direct oblique plane imaging can be considered as a detection PSF in other types of oblique plane microscopy using either beam scanning or selective-plane illumination if the same light clipping is involved. For such a system, a variety of illumination methods among point-, line-scanning, light-sheet, and so on could be coupled through either BS1 or BS1’ or other optical paths not shown in Fig. 1. The overall PSF of the imaging system then becomes the multiplication of the corresponding illumination and detection PSFs. 3. Theory and formulation Classical scalar diffraction theory simplified with the Fresnel approximation is only applicable to low NA or paraxial imaging systems [10]. It loses its validity in high NA systems (NA>0.6) which we mainly deal with for oblique plane microscopy. Scalar Debye theory [11], which is a more advanced version of scalar diffraction theory, does not use the paraxial approximation and considers an apodization factor of high aperture systems. However, it still neglects the vectorial nature of the light. Depolarization [11] in high NA imaging system influences on the PSF, making its main lobe broader along the incident polarization direction than that predicted by the scalar Debye theory. To accurately predict the performance of oblique plane imaging for any oblique angle and NA regimes, we adopt the vectorial diffraction theory [12] that considers the polarization of electromagnetic waves. 3.1. Vectorial diffraction theory
Fig. 2. Diffraction geometry. O: geometrical focus; X: an observation point; Y: a point on the exit pupil surface Σ where the incident field is refracted to E S . The distance YX is R ≡ rr− ' . Derived from a vector analogue of the Green’s second identity, the vectorial Kirchhoff integral [13,14] for a time-independent electric field E at an observation point r is expressed in SI unit as 1 Er() jGNˆ BNENEˆ 'G ˆ 'Gds , (1) = ∫∫ ω ()()×S +××∇S + ()⋅∇S 4π Σ where Σ is the wavefront surface over the exit pupil of an imaging system, Nˆ is a unit ray vector (normal to the wavefront), E S and BS are electric and magnetic fields respectively at the exit pupil, ω is the temporal frequency of the field, G is the Green function of a diverging spherical wave eRikR / with the distance R shown in Fig. 2, k is the wave number in medium, and ∇' is the gradient operator with respect to r ' . Applying Gauss’s law, vector identities and assuming k ≫ 1/R, Eq. (1) is reduced to
#207115 - $15.00 USD Received 25 Feb 2014; revised 8 Apr 2014; accepted 8 Apr 2014; published 1 May 2014 (C) 2014 OSA 5 May 2014 | Vol. 22, No. 9 | DOI:10.1364/OE.22.011140 | OPTICS EXPRESS 11143 −ik Er() G E ERNˆˆ NRE ˆˆ ds. (2) = ∫∫ SS−⋅()() +⋅ S 4π Σ In an aplanatic imaging system, parallel incident rays are assumed to refract at the spherical exit pupil towards the geometric focal point, making Nˆ and -r ' parallel in Fig. 2. 1 This aplanatic energy projection results in an angular apodization factor of cos2 θ [15]. Assuming that there is no change in polarization angles upon the refraction itself, the complex amplitude of the field E S is calculated on the geometric ground [16] as